MATEMATIQKI VESNIK

59 (2007), 47–56

UDK 517.521

originalni nauqni rad research paper

NORM AND LOWER BOUNDS OF OPERATORS ON WEIGHTED SEQUENCE SPACES

R. Lashkaripour and D. Foroutannia

Abstract. This paper is concerned with the problem of finding the upper and lower bounds of matrix operators from weighted sequence spaces l_{p}(v, I) into l_{p}(v, F). We consider certain matrix operators such as Cesàro, Copson and Hilbert which were recently considered in [7, 8, 11, 13] on the usual weighted sequence spaces l_{p}(v).

1. Introduction

We study the norm and lower bounds of certain matrix operators from l_{p}(v, I) into l_{p}(v, F) which were considered in [1, 2, 3, 4, 12] on l_{p} spaces and in [7, 8, 9, 10, 11, 13] on l_{p}(v) and Lorentz sequence spaces d(v, p), for certain matrix operators such as Cesàro, Copson and Hilbert operators.

If p ⩾ 1 and v = (v_{n}) is a decreasing non-negative sequence such that lim_{n→∞} v_{n} = 0 and ∑_{n=1}^{∞} v_{n} = ∞, we define the weighted sequence space l_{p}(v) as follows:

l_{p}(v) := { x = (x_{n}) : ∑_{n=1}^{∞} v_{n}|x_{n}|^{p} is finite },

with norm ∥ · ∥_{p,v} which is defined in the following way:

∥x∥_{p,v} = ( ∑_{n=1}^{∞} v_{n}|x_{n}|^{p})^{1/p}.

Let F be a partition of positive integers. If F = (F_{n}), where each F_{n} is a finite interval of positive integers and also max F_{n} < min F_{n+1} (n = 1, 2, …), we define the weighted sequence space l_{p}(v, F) as follows:

l_{p}(v, F) := { x = (x_{n}) : ∑_{n=1}^{∞} v_{n}|⟨x, F_{n}⟩|^{p} is finite },

where ⟨x, F_{n}⟩ = ∑_{j∈F_{n}} x_{j}. The norm of l_{p}(v, F) is denoted by ∥ · ∥_{p,v,F} and it is defined as follows:

∥x∥_{p,v,F} = ( ∑_{n=1}^{∞} v_{n}|⟨x, F_{n}⟩|^{p})^{1/p}.

AMS Subject Classification: 46A45, 40H05

For I_{n} = {n}, I = (I_{n}) is a partition of positive integers and l_{p}(v, I) = l_{p}(v) and ∥x∥_{p,v,I} = ∥x∥_{p,v}.

We write ∥A∥_{p,v,F} for the norm of A as an operator from l_{p}(v, I) into l_{p}(v, F), and ∥A∥_{p,v} for the norm of A as an operator from l_{p}(v) into itself.

We consider the lower bounds L of the form

∥Ax∥_{p,v,F} ⩾ L∥x∥_{p,v,I},

for all decreasing non-negative sequences x. The constant L is not depending on x. We seek the largest possible value of L, and denote the best lower bound by L_{p,v,F} for matrix operators from l_{p}(v, I) into l_{p}(v, F); it is denoted by L_{p,v}(A) when l_{p}(v, F) and l_{p}(v, I) are substituted by l_{p}(v).

The following statements give us some conditions adequate for the operators considered below, ensuring that ∥A∥_{p,v,F} is determined by decreasing, non-negative sequences.

(1) For all i, j, a_{i,j} ⩾ 0.

(2) For all subsets M, N of natural numbers having m, n elements respectively, we have

∑_{i∈M} ∑_{j∈N} a_{i,j} ⩽ ∑_{i=1}^{m} ∑_{j=1}^{n} a_{i,j}.

(3) ∑_{i=1}^{∞} v_{i} ∑_{j∈F_{i}} a_{j,1} is convergent.

If A = (a_{i,j}) is a matrix operator from l_{p}(v, I) into l_{p}(v, F) satisfying conditions (1) and (2), then decreasing, non-negative sequences are sufficient to determine the norm of A. Condition (3) ensure that at least finite sequences are mapped into l_{p}(v, F), see [6].

2. Upper bounds of matrix operators

The purpose of this section is to consider the norm of certain matrix operators from l_{p}(v, I) into l_{p}(v, F), the problem analogous to the one considered in [6, 8, 11, 13] on l_{p}(v) and Lorentz sequence spaces d(v, p) for certain matrix operators such as Cesàro, Copson and Hilbert operators.

Proposition 2.1. Let p ⩾ 1 and N ⩾ 1. Also, let F_{i} = {N_{i} − N + 1, N_{i} − N + 2, …, N_{i} − 1, N_{i}} and v_{n} = 1/n^{α}, where 0 < α ⩽ 1. If A is a bounded operator from l_{p}(v) into itself, then A is a bounded operator from l_{p}(v, I) into l_{p}(v, F) and also

∥A∥_{p,v,F} ⩽ N^{α/p}∥A∥_{p,v}.

Norm and lower bounds of operators … 49

Proof. Suppose x ∈ l_{p}(v, I) and y = Ax. We have
∥Ax∥_{p,v,F} = (∑_{i=1}^{∞} v_{i}|⟨y, F_{i}⟩|^{p})^{1/p}

= (∑_{i=1}^{∞} \frac{1}{i^{α}} |y_{N_{i}−N+1} + y_{n_{i}−N+2} + ⋯ + y_{N_{i}}|^{p})^{1/p}

⩽ (∑_{i=1}^{∞} \frac{|y_{N_{i}−N+1}|^{p}}{i^{α}} )^{1/p} + (∑_{i=1}^{∞} \frac{|y_{N_{i}−N+2}|^{p}}{i^{α}} )^{1/p} + ⋯ + (∑_{i=1}^{∞} \frac{|y_{N_{i}}|^{p}}{i^{α}} )^{1/p}

⩽ N^{α/p}( ∑_{k=1}^{∞} \frac{|y_{k}|^{p}}{k^{α}} )^{1/p} ⩽ N^{α/p}∥Ax∥_{p,v}.

This completes the proof.

In the following, we consider the above statement for p = 1.

Theorem 2.1. Suppose that A = (a_{i,j}) is a matrix operator satisfying conditions (1), (2) and (3). If sup U_{n}/V_{n} < ∞, where U_{n} = u_{1} + ⋯ + u_{n} and u_{n} = ∑_{i=1}^{∞} v_{i} ∑_{j∈F_{i}} a_{j,n} and V_{n} = v_{1} + ⋯ + v_{n}, then A is a bounded operator from l_{1}(v, I) into l_{1}(v, F) and

∥A∥_{1,v,F} = sup_{n} \frac{U_{n}}{V_{n}} .

Proof. Let x be in l_{1}(v, I) such that x_{1} ⩾ x_{2} ⩾ … ⩾ 0 and M = sup U_{n}/V_{n}. Then

∥Ax∥_{1,v,F} = ∑_{n=1}^{∞} v_{n}⟨Ax, F_{n}⟩ = ∑_{n=1}^{∞} v_{n}( ∑_{j∈F_{n}} ∑_{k=1}^{∞} a_{j,k}x_{k})

= ∑_{n=1}^{∞} u_{n}x_{n} = ∑_{n=1}^{∞} U_{n}(x_{n} − x_{n+1}).

Since ∥x∥_{1,v,I} = ∑_{n=1}^{∞} V_{n}(x_{n} − x_{n+1}), we have ∥Ax∥_{1,v,F} ⩽ M∥x∥_{1,v,I}. Therefore

∥A∥_{1,v,F} ⩽ M.  (I)

Further, we take x_{1} = x_{2} = ⋯ = x_{n} = 1 and x_{k} = 0 for all k ⩾ n + 1, then ∥x∥_{1,v,I} = V_{n}, ∥Ax∥_{1,v,F} = U_{n}. Hence

∥A∥_{1,v,F} ⩾ M.  (II)

Applying (I), (II) completes the proof of the theorem.

The Cesàro operator A is defined by y = Ax, where

y_{n} = \frac{1}{n}(x_{1} + x_{2} + ⋯ + x_{n}), for each n.

It is given by the Cesãro matrix a_{n,k} = \{ \begin{matrix}\frac{1}{n}, & for n ⩾ k, \\ 0, & for n < k. \end{matrix}

Theorem 2.2. Suppose that A is the Cesàro operator and p ⩾ 1. If N ⩾ 1 and F_{i} = {N_{i} −N +1, N_{i} −N +2, …, N_{i} −1, N_{i}} and v_{n} = 1/n^{α}, where 0 < α < 1, then A is a bounded operator from l_{p}(v, I) into l_{p}(v, F). Also, we have

∥A∥_{1,v,F} ⩽ N^{α}ζ(1 + α),

and

∥A∥_{p,v,F} ⩽ N^{α/p}p^{∗}, for p > 1 and p^{∗} = p/(p − 1).

Proof. Applying Proposition 5.1 of [8] and Proposition 2.1 the statement follows.

The Copson operator C is defined by y = Cx, where

y_{n} = ∑_{k=n}^{∞} \frac{x_{k}}{k} , for all n.

It is given as the transpose of the Cesàro operator: c_{n,k} = \{ \begin{matrix}\frac{1}{k} , & for n ⩽ k, \\ 0, & for n > k. \end{matrix} .

Firstly, we obtain the norm of Copson operator as the one from l_{1}(v, I) into l_{1}(v, F).

Lemma 2.1. ([8], Lemma 2.6) If α > 0, then

\frac{1}{n^{1−α}} ∑_{i=1}^{n} \frac{1}{i^{α}}

is decreasing with n and tends to 1/(1 − α) as n → ∞.

As an immediate consequence, we have

Theorem 2.3. Let C be the Copson operator. If F_{i} = {2i − 1, 2i} and v_{n} = 1/n^{α}, where 0 < α < 1, then C is a bounded operator from l_{1}(v, I) into l_{1}(v, F), and

∥C∥_{1,v,F} = \frac{2^{α}}{1 − α}.

Proof. With the above notation,

u_{n} = ∑_{i=1}^{∞} v_{i}(c_{2i−1,n} + c_{2i,n}).

Hence, \frac{u_{2n}}{v_{2n}} = \frac{2^{α}V_{n}}{n^{1−α}} and \frac{u_{2n−1}}{v_{2n−1}} = \frac{2V_{n−1} − v_{n}}{(2n − 1)v_{2n−1}} . Applying Lemma 2.1, we have

sup \frac{u_{2n}}{v_{2n}} = \frac{2^{α}}{1 − α} and also

\frac{u_{2n−1}}{v_{2n−1}} ⩽ \frac{u_{2(2n−1)}}{v_{2(2n−1)}} ⩽ sup_{n} \frac{u_{2n}}{v_{2n}} = \frac{2^{α}}{1 − α}.

Therefore ∥C∥_{1,v,F} = sup \frac{u_{n}}{v_{n}} = \frac{2^{α}}{1 − α}. This establishes the proof of the theorem.

In the following theorem a similar result is obtained for the Copson operator for p ⩾ 1.

Theorem 2.4. Let C be the Copson operator and p ⩾ 1. If N ⩾ 1 and F_{i} = {N_{i} − N + 1, N_{i} − N + 2, …, N_{i} − 1, N_{i}} and v_{n} = 1/n^{α}, where 0 < α < 1, then C is a bounded operator from l_{p}(v, I) into l_{p}(v, F). Moreover, we have

∥C∥_{p,v,F} = \frac{pN^{α/p}}{1 − α} .

Proof. Applying Theorem 4.2 of [8] and Proposition 2.1, we deduce that

∥C∥_{p,v,F} ⩽ \frac{pN^{α/p}}{1 − α} .

We now show that the reverse inequality holds. Choose ε > 0 and define r by α + rp = 1 + ε. Let y = Ax and x_{i} = 1/i^{r} for all n. Note that (x_{i}) is decreasing and x_{i} ∈ l_{p}(v, I). Then applying the integral estimate it follows that

y_{N_{i}−N+n} = ∑_{k=N_{i}−N+n}^{∞} \frac{1}{k^{1+r}} ⩾ \frac{1}{r(N_{i} − N + n)^{r}} ,

for all i and 1 ⩽ n ⩽ N. Therefore

∥Ax∥_{p,v,F} = (∑_{i=1}^{∞} v_{i}(y_{N_{i}−N+1} + y_{N_{i}−N+2} + ⋯ + y_{N_{i}−1} + y_{N_{i}})^{p})^{1/p}

⩾ (∑_{i=1}^{∞} v_{i}( ∑_{n=1}^{N} \frac{1}{r(N_{i} − N + n)^{r}} )^{p})^{1/p}

⩾ (∑_{i=1}^{∞} v_{i}( ∑_{n=1}^{N} \frac{N}{r(Ni)^{r}} )^{p})^{1/p} = \frac{N^{1−r}}{r} ∥x∥_{p,v,I}

= \frac{pN^{p−1+α/p}}{1 − α + ε} ∥x∥_{p,v,I} ⩾ \frac{pN^{α/p}}{1 − α + ε}∥x∥_{p,v,I}.

The statement of the theorem follows from the above inequality.

We recall that the Hilbert operator H is defined by the matrix

h_{i,j} = \frac{1}{i + j} , i, j = 1, 2, …

Let 0 < α < 1. As in the most studies on Hilbert operator, we use the well-known integral

∫_{0}^{∞} \frac{1}{t^{α}(t + c)} dt = \frac{π}{c^{α} sin απ} .

In the following, we are looking for an upper bound of the Hilbert matrix operator. At first, we consider the case p = 1 and give the exact solution for this problem.

Theorem 2.5. Let H be the Hilbert matrix operator, and F_{i} = {2i − 1, 2i} and v_{n} = 1/n^{α}, where 0 < α < 1. Then H is a bounded operator from l_{1}(v, I) into l_{1}(v, F), and also

∥H∥_{1,v,F} = \frac{2^{α}π}{sin απ} .

Proof. Using the usual notation we have

u_{n} = ∑_{i=1}^{∞} v_{i}(h_{2i−1,n} + h_{2i,n}) = ∑_{i=1}^{∞} \frac{1}{i^{α}} (\frac{1}{2i − 1 + n} + \frac{1}{2i + n}) .

Since u_{n} ⩾ ∫_{1}^{∞} \frac{1}{t^{α}} (\frac{1}{2t − 1 + n} + \frac{1}{2t + n}) dt and

∫_{0}^{∞} \frac{1}{t^{α}} (\frac{1}{2t − 1 + n} + \frac{1}{2t + n}) dt = \frac{π}{2^{1−α} sin απ} (\frac{1}{(n − 1)^{α}} + \frac{1}{n^{α}} ) ,

and also

∫_{0}^{1} \frac{1}{t^{α}} (\frac{1}{2t − 1 + n} + \frac{1}{2t + n}) dt ⩽ \frac{2}{n(1 − α)},

we have

n^{α}u_{n} ⩾ \frac{π}{2^{1−α} sin απ} (1 + \frac{n^{α}}{(n − 1)^{α}} ) − \frac{2n^{α}}{n(1 − α)} ⩾ \frac{2^{α}π}{sin απ} − \frac{2}{n^{1−α}(1 − α)}.

Therefore

∥H∥_{1,v,F} = sup_{n} n^{α}u_{n} ⩾ \frac{2^{α}π}{sin απ} .

It was shown in [6] that ∥H∥_{1,v} = \frac{π}{sin απ}, and so applying Proposition 2.1. we have

∥H∥_{1,v,F} ⩽ \frac{2^{α}π}{sin απ} .

This establishes the proof of the theorem.

We now consider the norm of the Hilbert matrix operator for the general case, when p > 1.

Theorem 2.6. Suppose that H is the Hilbert matrix operator and p > 1. If N ⩾ 1 and F_{i} = {N_{i} − N + 1, N_{i} − N + 2, …, N_{i} − 1, N_{i}} and v_{n} = 1/n^{α}, where 1 − p < α < 1, then H is a bounded operator from l_{p}(v, I) into l_{p}(v, F). Moreover, we have

∥H∥_{p,v,F} = \frac{πN^{α/p}}{sin[(1 − α)π/p]}.

Proof. Applying Theorem 3.2 in [8] and Proposition 2.1, we deduce that

∥H∥_{p,v,F} ⩽ \frac{πN^{α/p}}{sin[(1 − α)π/p]}.

To show the reverse inequality, take r = (1 − α)/p, so that α + rp = 1. Fix M, and let

x_{j} = \{ \begin{matrix}\frac{1}{j^{r}} , & for j ⩽ M, \\ 0, & for j > M. \end{matrix}

Then (x_{j}) is decreasing and ∑_{j=1}^{∞} v_{j}x_{j} = ∑_{j=1}^{M} \frac{1}{j}. Also, let y = Hx. By routine methods (we omit the details), one finds that

∑_{i=1}^{n} v_{i}(y_{N_{i}−N+1} + y_{N_{i}−N+2} + ⋯ + y_{N_{i}−1} + y_{N_{i}})^{p} ⩾

⩾ \frac{πN^{1−r}}{sin rπ} ∑_{i=1}^{M} \frac{1}{i} − g(r) ⩾ \frac{πN^{α/p}}{sin[(1 − α)π/p]} ∑_{i=1}^{M} \frac{1}{i} − g(r),

where g(r) is independent of M. Clearly, the required statement follows.

3. Lower bounds of matrix operators

In this part of the study we are looking for the lower bounds of matrix operators considered in Section 2.

Let p ⩾ 1 and A be a matrix operator with non-negative entries. If y = Ax and v is a decreasing sequence, then for each non-negative sequence x, we have

∥Ax∥_{p,v,F}^{p} = ∑_{i=1}^{∞} v_{i}( ∑_{j∈F_{i}} y_{j})^{p} ⩾ ∑_{i=1}^{∞} v_{i} ∑_{j∈F_{i}} y_{j}^{p}

⩾ ∑_{i=1}^{∞} v_{i}y_{i}^{p} = ∥Ax∥_{p,v}^{p}.

It follows that L_{p,v,F} (A) ⩾ L_{p,v}(A).

Corollary 3.1. Suppose that A is the Cesàro operator and p ⩾ 1. If v_{n} = 1/n, then L_{p,v,F} (A) ⩾ 1.

Proof. If we apply Theorem 4 in [7], we deduce that L_{p,v}(A) = 1 and so we have the statement.

The Copson matrix is an upper triangular matrix. We will solve the lower bound problem through the next statement. In fact, we characterize a class of operators for which the lower bound constant is equal to one.

Theorem 3.1. Suppose that A is an upper triangular matrix, i.e. a_{n,k} = 0 for n > k, and ∑_{n=1}^{k} a_{n,k} = 1 for all k (in other words, A is a quasi-summable matrix). Let p ⩾ 1 and v = (v_{n}) be a non-negative decreasing sequence. Then L_{p,v,F} (A) = 1.

Proof. If we apply Proposition 2 in [7], we have L_{p,v}(A) = 1. Hence L_{p,v,F} (A) ⩾ L_{p,v}(A) = 1. Since 1 ∈ F_{1} and Ae_{1} = e_{1}, we deduce that

∥Ae_{1}∥_{p,v,F} = ∥e_{1}∥_{p,v,I} = v_{1}.

This completes the proof of the theorem.

We now generalize Theorem 1 of [7] for certain matrix operators from l_{p}(v, I) into l_{p}(v, F) and deduce the lower bound for the Hilbert matrix operator.

Lemma 3.1. [7] Let p ⩾ 1. Suppose that (a_{j}), (x_{j}) are non-negative sequences and (x_{j}) is decreasing and tends to 0. Write A_{n} = ∑_{j=1}^{n} a_{j} (with A_{0} = 0), and B_{n} = ∑_{j=1}^{n} a_{j}x_{j}. Then:

(i) B_{n}^{p} − B_{n−1}^{p} ⩾ (A_{n}^{p} − A_{n−1}^{p})x_{n}^{p} for all n; (ii) if ∑_{j=1}^{∞} a_{j}x_{j} is convergent, then

( ∑_{j=1}^{∞} a_{j}x_{j})^{p} ⩾ ∑_{n=1}^{∞} A_{n}^{p}(x_{n}^{p} − x_{n+1}^{p}).

Corollary 3.2. If (x_{j}) is a non-negative decreasing sequence and X_{n} = x_{1} + ⋯ + x_{n}, then for each n, X_{n}^{p} − X_{n−1}^{p} ⩾ [n^{p} − (n − 1)^{p}]x_{n}^{p}.

Theorem 3.2. Suppose that p ⩾ 1 and A = (a_{i,j}) is a matrix operator from l_{p}(v, I) into l_{p}(v, F) with non-negative entries. Write r_{j,i} = ∑_{k=1}^{i} a_{j,k} and

S_{i} = ∑_{n=1}^{∞} v_{n}( ∑_{j∈F_{n}} r_{j,i})^{p}.

Then L_{p,v,F}^{p} (A) = inf_{n} \frac{S_{n}}{V_{n}}.

Proof. Let x be in l_{p}(v, I) such that x_{1} ⩾ x_{2} … ⩾ 0 and m = inf S_{n}/V_{n}. Applying Lemma 3.1, we have y_{i}^{p} ⩾ ∑_{n=1}^{∞} r_{i,n}^{p}(x_{n}^{p} − x_{n+1}^{p}). Hence

∥Ax∥_{p,v,F}^{p} = ∑_{n=1}^{∞} v_{n}( ∑_{j∈F_{n}} y_{j})^{p} ⩾ ∑_{n=1}^{∞} v_{n} ∑_{i=1}^{∞}( ∑_{j∈F_{n}} r_{j,i})^{p}(x_{i}^{p} − x_{i+1}^{p})

= ∑_{i=1}^{∞} S_{i}(x_{i}^{p} − x_{i+1}^{p}).

Since ∥x∥_{p,v,I}^{p} = ∑_{n=1}^{∞} V_{n}(x_{n}^{p} − x_{n+1}^{p}), we deduce that

∥Ax∥_{p,v,F}^{p} ⩾ m∥x∥_{p,v,I}^{p}.

Therefore L_{p,v,F} (A) ⩾ m.

Further, we take x_{1} = x_{2} = ⋯ = x_{n} = 1 and x_{k} = 0 for all k ⩾ n + 1; then ∥x∥_{p,v,I}^{p} = V_{n} and ∥Ax∥_{p,v,F}^{p} = S_{n}. Hence

L_{p,v,F} (A) ⩽ m.

This establishes the proof of the theorem.

Note 3.1. For p > 1, the last part of Theorem 3.2 shows that ∥A∥_{p,v,F}^{p} ⩾ sup_{n} S_{n}/V_{n}, but l_{p}(v, F) = l_{p}(v) when F_{i} = {i} and equality does not hold (see [8]). Write

t_{n} = ∑_{i=1}^{∞} v_{i}( ∑_{j∈F_{i}} a_{j,n})^{p},

and

s_{n} = S_{n} − S_{n−1} = ∑_{i=1}^{∞} v_{i}[( ∑_{j∈F_{i}} r_{j,n})^{p} − ( ∑_{j∈F_{i}} r_{j,n−1})^{p}],

where S_{n} = s_{1} + ⋯ + s_{n}. For p = 1, we have t_{n} = s_{n}. It is elementary that inf_{n}(S_{n}/V_{n}) ⩾ inf_{n}(s_{n}/v_{n}). We now apply Lemma 3.1 to deduce the following result.

Proposition 3.1. If A satisfies all conditions mentioned in Theorem 3.1 and (a_{i,j}) decreases with j for each i, then

L_{p,v,F} (A)^{p} ⩾ inf_{n}[n^{p} − (n − 1)^{p}]\frac{t_{n}}{v_{n}} .

Proof. It follows from Corollary 3.2 that

( ∑_{j∈F_{i}} r_{j,n})^{p} − ( ∑_{j∈F_{i}} r_{j,n−1})^{p} ⩾ [n^{p} − (n − 1)^{p}]( ∑_{j∈F_{i}} a_{j,n})^{p}.

Thus

s_{n} ⩾ [n^{p} − (n − 1)^{p}] ∑_{i=1}^{∞}( ∑_{j∈F_{i}} a_{j,n})^{p} = [n^{p} − (n − 1)^{p}] t_{n}

and so we have the statement.

In the following statement we consider the lower bound constant for the Hilbert operator H.

Theorem 3.3. Suppose that H is the Hilbert operator, and p ⩾ 1. Let F_{i} = {2i − 1, 2i} and v_{n} = 1/n^{α}, where 0 < α < 1. Then

L_{p,v,F} (H)^{p} ⩾ ∑_{k=1}^{∞} \frac{1}{k^{α}(k + 1/2)^{p}} .

Proof. Let E_{k} = {i ∈ Z : (k − 1)n < i ⩽ kn}, where k ⩾ 1. If i ∈ E_{k}, then (\frac{i}{n})^{α}(2i + n)^{p} ⩽ k^{α}(2kn + n)^{p}. Since E_{k} has n members,

n^{p+α−1} ∑_{i∈E_{k}} \frac{1}{i^{α}(2i + n)^{p}} ⩾ \frac{n^{p}}{k^{α}(2kn + n)^{p}} = \frac{1}{k^{α}(2k + 1)^{p}} .

Hence

n^{p+α−1} ∑_{k=1}^{∞} \frac{1}{k^{α}(2k + n)^{p}} ⩾ ∑_{k=1}^{∞} \frac{1}{k^{α}(2k + 1)^{p}} ,

and also

inf_{n} n^{p+α−1} ∑_{k=1}^{∞} \frac{1}{k^{α}(2k + n)^{p}} = ∑_{k=1}^{∞} \frac{1}{k^{α}(2k + 1)^{p}} .

We now apply Proposition 3.1 and with the above notation,

t_{n} = ∑_{i=1}^{∞} \frac{1}{i^{α}} (\frac{1}{2i − 1 + n} + \frac{1}{2i + n})^{p} ,

and L_{p,v,F} (H)^{p} ⩾ inf_{n}[n^{p} − (n − 1)^{p}]n^{α}t_{n}.

Since n^{p} − (n − 1)^{p} ⩾ n^{p−1}, we have

L_{p,v,F} (H)^{p} ⩾ inf_{n} n^{p+α−1} ∑_{i=1}^{∞} \frac{1}{i^{α}} (\frac{1}{2i − 1 + n} + \frac{1}{2i + n})^{p}

⩾ 2^{p} inf_{n} n^{p+α−1} ∑_{i=1}^{∞} \frac{1}{k^{α}(2k + n)^{p}} = 2^{p} ∑_{k=1}^{∞} \frac{1}{k^{α}(2k + 1)^{p}} .

This completes the proof of the theorem.

As mentioned in Theorem 3 in [7], we have

L_{p,v}(H)^{p} = ∑_{k=1}^{∞} \frac{1}{k^{α}(k + 1)^{p}} .

Therefore we have shown that L_{p,v,F} (H) > L_{p,v}(H).

REFERENCES

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(received 22.10.2006)

Dept. of Math., Sistan and Baluchestan University, Zahedan, Iran

E-mail: lashkari@hamoon.usb.ac.ir, d foroutan@math.com